Lucas Numbers
Number Theory
Level
3
Let L(x) be a sequence, such that L(1)=1, L(2)=3, and L(n+2) = L(n+1) + L(n). Which of the following equals L(α)? ( nearℤ is nearest integer function)
L(nearℤ(α↑π))
nearℤ(π^n)
nearℤ(e^n)
nearℤ(φ^(n-1))
nearℤ(eπ^n)
L(α-1) + L(a-2)
nearℤ(α^(n/2))
nearℤ(φ^n)
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1 solution
One method is by doing induction on L(n) by assuming it is verified and true for L(n-1) and L(n-2), second method is by observation method from Fibonacci, i.e, establish a formula between two terms of L(n) and some terms of F(n), and other methods if possible, will be appreciated by me, as I am looking for easier solutions. I did not post detailed solutions because by not giving total solutions, the solver's curiosity is provoken and hence is not boring compared to detailed solutions.